Physics 263: Problem Set #11

1. (BTM 7.1.1.) The key to showing equations (7.1.2) and (7.1.3) geometrically is drawing a careful sketch. Draw a large version of the left graph in Figure 7.1, and mark A_x, A_y, A'_x, and A'_y by drawing lines perpendicular to the various axes from the end of the vector. Draw more lines to make right triangles where A_x and A_y are the hypoteneuses (hypoteni?). You'll note many similar triangles with the same angle theta. You should also be able to find that A'_x is split into two pieces as indicated by (7.1.2). For the second part, just carry out the squares and use the standard trigonometric identity for the sum of sin squared and cos squared.
2. (BTM 7.2.2.) Recall that "orthogonal" means that they are at right angles, which also means that the dot product is zero. To carry out the cross products, I advise using Equation (7.1.29) and the others; this will be useful for future discussions. You can check your answers using MATLAB's dot and cross functions.
3. (BTM 7.2.3.) I'm not sure "triple product" is actually defined in the text (maybe in problem 7.1.3), but in any case, it means a . (b x c) ["a dot b cross c"].
4. (BTM 7.2.4.) To make your sketch, select some values of omega*t equal to standard angles (like pi/4, pi/2, pi, etc.). The motion is pretty obvious in polar coordinates.
5. (BTM 7.1.3.) You can use the fact discussed in the paragraph before this problem relating the cross product of two vectors to the area of the parallelogram with sides given by the vectors.